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Conditions for an operator to be bounded

Let $E$ be a Banach space, and let $T : E \to E^*$ be a linear operator. In each of the following cases, prove that $T$ is bounded.
a) $\langle Tx,x \rangle \geq 0,\ \forall x \in E$;
b) $\langle Tx,y \rangle=\langle Ty,x\rangle,\ \forall x,y \in E$.

Semnificatia e urmatoarea, pentru o functionala $f \in E^*$ se noteaza $f(x)=\langle f, x \rangle$.